How Can I Spread Crepe Batter Before It Cooks?

Your batter is too thick. You say it's refrigerated, but if that's the case it's likely thicker than if it was at room temperature. Also, as it sits it gets thicker, and if you're making a solid batch, for a few people, you need to keep thinning it as you go. Take the liquid you were using to begin with (milk), and add very small quantities at a time, on the order of tablespoons, stirring and evaluating the texture after each one. If you go too far, you'd have to add flour again and get into an overly involved cycle of correction.

Crepe batter thickening is a common problem in crepe establishments in the US. The crepes early in the day are fine but they get thicker and thicker as the day progresses because nobody thinks to thin the batter.

It sounds like maybe there's a couple of possibilities.

1. The pan is too hot. This can happen even over lower heat settings if you leave the pan to preheat for too long, or it may be that the burner's "medium-low" setting is just too hot, and you should use an even lower setting.
2. The batter consistency is wrong. Crepe batter should be very thin, if it is cooking before you can spread it out that indicates that perhaps the batter is too thick.
3. You aren't using enough batter. If you're satisfied with the thickness of the crepes you're getting, then you need to use more batter in order to cover the rest of the pan. You don't mention the thickness in your question, but since you're still calling them crepes and not pancakes, I'm guessing they're thin enough?

I think you're having either some combination of the first two issues or the third issue (most likely on its own, but possibly combined with #1).

You mention crispy edges, which sounds like the first issue, but you also mention it rapidly becoming "too thick to flow" so I think you may have a little of both of the first two going on. Given you're getting crispy edges and consider that a problem (every good crepe I've ever had has had lightly crispy edges, so I wouldn't necessarily call that a problem), it is possible that your batter is too thin, but I think that's unlikely-- crepe batter is better too thin than too thick, in my experience.

You can actually buy tools which help to spread batter. My sister got me a hot-plate crepe maker for Christmas the other year, which came with one of these:

Which I find does the job perfectly! You should be able to find one online pretty easily or DIY one yourself.

https://www.youtube.com/watch?v=uhKFAlk-gtU

First, this can indeed be a thick batter, as the other answers mentioned. I would recommend using Ruhlman's ratio of 1:2:2 flour to milk to egg as a starting point of what a good crepe batter should be like. You can experiment with other recipes if you want something nonstandard, but first do some batches to get a feel for the proper consistency. And don't eyeball, use a scale.

Second, the swirl method produces slightly thicker crepes, like Swabian Flädle. If you want a true thin French crepe, you should use a T shaped tool for spreading the batter. It does need some practice, but it's learnable in a reasonable time. If you instead stay with the swirl method, even with the proper consistency, the bottom layer of the batter will bake before you've swirled the top, so you need sufficient batter to allow for that, as senschen said.

The lacy edge is also common with swirling, although proper consistency reduced it too. It also depends on good swirling technique, if you can hit an angle where the batter doesn't climb the pan wall, you don't get the lace. What makes it worse are pans with sloping sides, instead of straight ones (less hot) or real crepe pans without walls. Also, gas makes it worse, because it warms the pan sides. Again, if you want your crepes to be that close to the original ( thickness, no lacy edge, proper Browning, etc) you should use proper tools and not a random pan without a spreader.

Higher heat, thinner batter, and don't put it in the fridge. No tools necessary. I recommend thin non-stick pans, and a gas stove.

The batter should be thin enough it basically spreads itself, as soon as it hits a pan hot enough to totally 'melt' it. The only reason to swirl it is to circularize it. It should be a consistency that you could drink without needing a glass of water afterwards.

Paper thin browned edges is the perfect crape IMO. That's when you flip it. I don't have mom's recipe atm but it's probably a little heavy on the butter. Real butter, with which you coat each pan liberally (use more than one pan or you'll be there all day) in between each cake and allow it to come back up to temp. If the pan isn't hot enough it'll just make regular pancakes, even if you've got the batter right.